We present a mechanism by which circularly driven phonon modes in the rare-earth trihalides generate giant effective magnetic fields acting on the paramagnetic 4f spins. With cerium trichloride (CeCl₃) as our model system, we calculate the coherent phonon dynamics in response to the excitation by an ultrashort terahertz pulse using a combination of first-principles calculations and phenomenological modeling. We find that effective magnetic fields of over 100 T can possibly be generated that polarize the spins for experimentally accessible pulse energies. This mechanism potentially creates a way to control the magnetic and electrical order of ferromagnets and ferroelectrics through interfacial coupling with the phonon-induced magnetization in heterostructures.